If you have four sticks, you can make a whole ton of different quadrilaterals, because the angles between the sticks can change. But, if you have three sticks, they will make only one triangle. Those angles can’t change, because if they do, the sticks won’t come together anymore. So, a triangle is very resistant to deformation, which is another way of saying it’s very strong.

They aren’t.

Hexagons are, a la honeycomb. A quick YouTube search will yield some cool videos about the geometric reasons for this, with practical examples.

The universe itself seems to prefer spheres. Spheres have maximum surface area to volume ratio.

Each shape has properties. It is a fun rabbit hole to go down if you care to learn about the golden ratio, Pyramids, resonance frequencies, Greek dudes using sun dials to calculate the size of the earth, navigation by constellation, on and on.

Think of each angle in your shape as a hinge with no resistance. Each side is a fixed length.

A square is just a parallelogram with perpendicular sides. Those same 4 sides can be used to make any parallelogram with the same sides. There is no “invalid” point where these four sides won’t make a parallelogram, only a gradient of changing angles. It only becomes a square when the angles are all 90 degrees. That means if you push on one corner of your square, all four hinges will bend and your square becomes a different parallelogram with the same sides.

A triangle has 3 sides. Those three sides form three angles with a total of 180 degrees. For three sides of a fixed length, you can only ever make a triangle with the same angles. All the angles in between are invalid. That means when you push on the corner of your triangle, there is no gradient and your hinges don’t bend.

This is why a triangle is a strong shape. A square can be bent, a triangle has to be broken, and every useful building material on earth is harder to break than bend.

Places where two long items join are typically weak spots due to leverage. Try holding two pencils ends together at a ninety degree angle with your hands. Someone pushing on the other end doesn’t need much force at all to change that angle despite your strength because they are gaining tremendous leverage by pushing against the far end of the pencil.

A triangle makes that far end impossible to move. Thus the whole shape doesn’t change.

Triangles are typically the strongest shape to use in a frame, when loads are applied to the node points, in a direction that is not perpendicular to the plane of the frame.

This is because in order for a node to deflect the members that connect to it must stretch and compress, this is in comparison to shapes with more sides where the sides can stay the same length while the members bend.

Because of how the math works out for the structural properties, the strength and rigidity of virtually any section (shape of the pieces you’re making the frame out of) is higher in tension/compression than in bending. So if you can force all pieces to be in tension/compression rather than in bending you’ll end up with a stronger frame.

In other applications triangles are not the strongest shape – examples are arches and suspension structures where the ideal shape is typically somewhere between a catenary and a parabolic curve (depending on load distribution) and for shapes intended to contain pressure (tanks) a cylinder/sphere is the strongest- in both cases because for the loads applied they force all forces to be in tension and compression.

As already noted it is a bit of an oversimplification to say “triangles are the strongest”. However for many materials their ability to resist compression or tension is much greater than their ability to resist bending.

For example if you wanted to break a pencil how would you do it? You would probably try to bend it in the middle until it breaks, not try to smash it from either end or just pull it straight apart.

What arranging such materials in a triangle does is make it so in order to bend one leg of the triangle you would need to either lengthen or compress at least one of the other two legs (or detach the corners, but we can ignore that). The relatively weak resistance to bending is replaced by resistance to compression or stretching.

Strongest in what way? And for what purpose?

There are strengths to different shapes, arrangements, and so on, but it depends a lot on the context. Maximizing volume to surface area gives you a sphere. Intersecting numerous cells that gives maximum strength for minimum material will give you hexagon. Maximizing compressive resistance gives you an arch or a circle. A lot is just in how force travels through the shape, and from what angle/origin. Triangles are pretty good in distributing force if they’re not delivered to an edge, but instead to one of the corners. But then, I wouldn’t say they are universally the strongest–not by a long shot.

Because for 3 given sidelength there are only 3 specific angles at wich they can be, so there is no way to turn the points where they are joined (usually weakpoints)

Compare that to a square. If you push from the side then the angles can all simultanously shift to form a rhombus. Add a diagonal extra connection in the middle and it’s suddenly impossible

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